223 problems

Fix $d\ge 2$. Let $\eta=(\eta(x))_{x\in\mathbb{Z}^d}$ be an i.i.d. initial mass configuration for the divisible sandpile, with mean density $\rho$, and let…

For every infinite, locally finite, transitive graph $G=(V,E)$, let $\mathbb{P}_p$ denote Bernoulli bond percolation with parameter $p$, let $o\in V$, and define…

Higher-dimensional max-flow min-cut conjecture. There exists a map $\mu$ on the unit sphere such that, for every convex set $A$ with $0$ in its interior,

Let $G=(V,E)$ be a finite connected graph and let $T\subset V$. The bunkbed graph $G_T$ has vertex set $V\times\{0,1\}$, two horizontal copies of every edge of $G$, and a vertical…

Tile the unit square by, possibly infinitely many, squares of varying sizes, with at most three squares meeting at any corner. Color each square black or white independently, each…

Critical-percolation conjecture.

Let $G$ be a finite graph, let $\lambda(G)$ denote the spectral radius of its adjacency matrix, and let $G_p$ be obtained from $G$ by retaining each edge independently with probabi…

Let $G$ be an infinite, connected, transitive, simple graph of vertex degree $d$ that is not one-dimensional, and let $p_c(G)$ be its Bernoulli bond percolation critical probabilit…

Let $G=(V,E)$ be an undirected graph, and let $(V^-,E^-)$ and $(V^+,E^+)$ be two copies of $G$. For each $v\in V$, write $v^-$ and $v^+$ for its copies, and form the bunkbed graph…

Random-current sharp-transition conjecture. The single random current on $\mathbb{Z}^d$ has a unique sharp percolative phase transition, namely

Let $G$ be a bounded-degree plane triangulation. Write $G^{1/2}$ for site percolation on $G$ in which each vertex is retained independently with probability $1/2$. The graph $G$ is…

Let $q_c:[0,1]\to[0,1]$ be the critical function for anisotropic bond percolation, and let $p_c(d)$ denote the critical parameter for $d$-dimensional percolation. Write…

Schramm's locality conjecture. Then

Strict $q$-to-$q$ threshold conjecture. For every $q\in(1,\infty)$,

Let $d\geq 2$ and $q\geq 1$. For the random-cluster model on $\mathbb{L}^d$, let $\phi_{p,q}^0$ and $\phi_{p,q}^1$ be the infinite-volume random-cluster measures with free and wire…

Let $D$ be a simply connected domain in the complex plane, and let $a,b,c,x$ be boundary points in counterclockwise order. Suppose that $D$ is conformally equivalent to the equilat…

The bunkbed conjecture for transitive tournaments. The statement

Consider the infinite connected components of the corner-percolation configuration, each assigned its integer-valued level. Level-bijection conjecture. The function which maps each…

Let $(G_n)_n=(V_n,E_n)_n$ be a sequence of finite, connected, vertex-transitive graphs with vertex degree $\bDelta_n$ diverging as $n\to\infty$. For a fixed sequence and fixed $u$,…

Monotonicity conjecture. For $\lambda<\lambda'\in(0,1]$,

Häggström–Pemantle's conjecture. In every dimension $d\geq 2$, infinite coexistence has positive probability if and only if the infection intensities are equal:

Let $ell geqslant 2$ and let ${\mathcal R}$ be an $ell$-vertex size rule. Define $t_{\mathrm{b}}^{{\mathcal R}}$ as the supremum of the set of $t \geqslant 0$ for which the suscept…

Let $G$ and $G'$ be groups with Cayley graphs whose asymptotic cones are isometric. A group is non-elementary hyperbolic if it is hyperbolic and is not virtually cyclic. Sapir's co…

Commuting infinite subgroups conjecture. The group $G$ contains two infinite subgroups $H_1$ and $H_2$ which commute with each other:

Consider the competition process on $\mathbb{Z}^d$ with blue and yellow infections, whose transmission parameters are $p_{\text{b}}$ and $p_{\text{y}}$, respectively. Let $p_c$ be…